What is the stochastic differential equation that defines geometric Brownian motion?

Q: What is the stochastic differential equation that defines geometric Brownian motion?

A stochastic process $S$ is defined as following geometric Brownian motion if it satisfies the stochastic differential equation $dS = \mu S\,dt + \sigma S\,dW$, where $\mu$ and $\sigma$ are constants representing the drift and volatility, and $W$ is a standard Brownian motion.

A stochastic process

S

is defined as following geometric Brownian motion if it satisfies the stochastic differential equation

dS = \mu S\,dt + \sigma S\,dW

, where

\mu

and

\sigma

are constants representing the drift and volatility, and

W

is a standard Brownian motion.

Conditions

$\mu$ and $\sigma$ are constants.
$W$ is a standard Brownian motion.

Reasoning, step by step

Identify the stochastic process $S$ .
State the differential equation $dS = \mu S\,dt + \sigma S\,dW$ .
Specify that $\mu$ and $\sigma$ are constant parameters.
Identify $W$ as the standard Brownian motion driving the equation.

Example

The instructor writes the formal definition on the board: 'We say that a process S follows geometric Brownian motion if it satisfies the SDE: dS = $\mu S dt + \sigma S$ dW for $\mu$ and $\sigma$ constant.'

Common misconceptions

Assuming $\mu$ and $\sigma$ can vary with time or the state of the process.
Confusing geometric Brownian motion with standard Brownian motion, which lacks the proportional $S$ terms.

Watch the explanation

YouTubeGeometric Brownian Motion

0:16 – 1:12Watch this moment ↗

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